Here's an old paradox where the rationalist might not have the upper hand . What do you guys think? Of course there's a good chance you've read this one before...if so share your thoughts.

An alien comes from many stars away and explains to you that he has the power to predict how you'll make decisions. You are convinced based upon seeing him correctly predict other humans' decisions that he can do this. You're not absolutely 100% sure, but you are 99.99% sure, as sure as we can be of anything in this life.

With this in mind, the alien sets up the following experiment: he gives you two sealed boxes, called A and B. Inside A is $100, and that is a sure thing. Inside of B is either nothing or $1,000,000. The alien says that you are free to take either just box B or to take both boxes. The catch is that he has already made his prediction about which choice you'll make. If he predicted that you'll only take B then he generously stuffed it with the one million (he tells you this). If he predicted that you'll take both boxes then he put nothing in B. After leaving the boxes, the alien departs in his spaceship never to return, leaving you to ponder this decision. Do you take both of the boxes? Or only box B?

This riddle is terribly interesting to me because it is one of the few scenarios where rigidly rational decisions making leads to a less optimal outcome than more "stupid" decision making. But isn't that a contradiction in terms? To me, this paradox illustrates the limits of rationality, or maybe that we just think about rationality in the wrong way. to choose to take either just B or to take A and B.

Isn't it rational thinking that tells you to go for just box B because then he'll have put $1,000,000 in it?

Or is that part (under which circumstance he'll have put $1,000,000 in box B ) not known to the decision maker? In that case, the problem is insufficient knowledge - same as I offer you $1,000 for free but I'm not telling you that I'm going to shoot you if you take it - the rational decision is to accept the free money

Point being that any rational decision needs to be an informed decision, meaning that you have checked the facts and know as much as possible about your chance for success, resulting advantages, their likelihood and the negative side, risks and the likelihood of them occurring in case of failure as well - and being aware of what you weren't able to find out. In case of the alien guy, you might try to find out if he has the habit of tricking people and therefore rationally arrive at the conclusion to take box B. In case of the guy with $1,000, the line of dead bodies with $1,000 notes in their hands might give a clue

Did the alien actually tell me "If he predicted that you'll only take B then he generously stuffed it with the one million." or is that an assumption I make, because it seems like a big assumption. If he has already 'predicted' it, it doesn't matter if you take both (Which could be a way to make his prediction true), to destroy any prediction you could take niether. However if the aforementioned assumption wasn't an assumption you would take B.

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Who cares about $100 when there's $1,000,000 in the other box. The pay off for taking box B far exceeds that of box A and if the alien is wrong (unlikely it seems) well your only losing $100 (well your not really losing anything but...). Given that the alien just left the boxes and flew off in his spaceship before you took the boxes, then its all a bit dull anyway.

Take b, if there's nothing in it then take A too, thats the logical choice (you end up taking both and only get $100, alien is correct). What other choice is there really, take b, find theres nothing in it then leave A with $100 just sitting there?

I suppose it depends if you care more for $100 than you do for making the alien wrong .

Rationality is not the same as perfection and does not necessarily lead to optimal solutions at all times. A more interesting (and realistic) scenario is the Prisoner's dilemma, where rational behavior generally leads to suboptimal results.

I would think the OP's story is rather easy to solve: assuming the alien is speaking the truth, one choice gets you one million, the other one hundred. If you take into account probabilities, you get for B = 0.9999 * $1,000,000 + 0.0001 * $1,000,000= $1,000,000 and A+B = 0.9999 * $100 + 0.0001 * $1,000,100 = $200, hence a rational actor still picks B. The only irrational part I can see is in believing the alien can predict your future, but you assign percentages of probability to that, so that makes it moot.

rigidly rational decisions making leads to a less optimal outcome than more "stupid" decision making.

Rational does not imply optimal nor does it lead to winning strategy.

Plenty of cases demonstrate that rational approach for selection of strategy will frequently be sub-optimal and that many day-to-day events are result in better outcomes using irrational thinking. Especially when it comes to society, where crowd behavior (crowdsourcing) can result in better outcome.

Perhaps a better term would counter-intuitive.

Then there's Parrondo's paradox, where choosing losing strategies results in winning. Thinking of rationality by involving an Oracle is only useful in theory, in practice there are too many unknowns involved.

For me, the obvious conclusion is that the alien is judging humanity by examining whether I am greedy or not. Will they destroy us if we are greedy? Will they establish more general contact if we take the one box? What if they respect greed, rather than co-operation?

Which box to take is now not a matter of 100 dollars, but the fate of the world. Think outside the box.

This is not a paradox at all. Given that the aliens can accurately predict what I will do, then I am living in a wholly deterministic universe, where the future is just as fixed as the past. As humans, we are used to seeing the future as something that we have control over, but not the past. Under the assumption that the universe is predictable, there is no difference except in our perception. Our memory of the past is pretty good, but still a bit "fuzzy." Our own "memory" of the future (i.e. our ability to extrapolate) is significantly more fuzzy, but perhaps the alien's mind is structured in some highly advanced/different way such that it does a better job of predicting/remembering the future than it does of knowing the past. In a deterministic universe there is no reason why this should not be so, given the correct information and apparatus.

The nuances of this are difficult to imagine given our own condition and the way the universe seems to work, e.g. to the alien setting an object down does not necessarily make sense the way it does to us, but the main point stands: if the alien really has perfect knowledge of the future, from his perspective, he might as well remember having seen us already make the choice. I couldn't expect a person, "rational" or otherwise, to change what they have already done in the past in order to attain some goal, because it's not possible. But, from the perspective of the alien, this is what's happening. It's only from your own perspective that there is a "choice."

A better question is whether such a thing is even possible; can I make a prediction about what you'll do, tell you what you're going to do, and expect you to do it? Even in a deterministic universe, intuitively, it might not be possible for such a prediction to actually exist, since the prediction, if it's a normal sort of computational model, will need to account for its own existence, which means that the prediction has an information content equal to stuff about the universe plus its own information content (i.e. x = x + c). I don't know about this one.

Rationality is not the same as perfection and does not necessarily lead to optimal solutions at all times. A more interesting (and realistic) scenario is the Prisoner's dilemma, where rational behavior generally leads to suboptimal results.

I would think the OP's story is rather easy to solve: assuming the alien is speaking the truth, one choice gets you one million, the other one hundred. If you take into account probabilities, you get for B = 0.9999 * $1,000,000 + 0.0001 * $1,000,000= $1,000,000 and A+B = 0.9999 * $100 + 0.0001 * $1,000,100 = $200, hence a rational actor still picks B. The only irrational part I can see is in believing the alien can predict your future, but you assign percentages of probability to that, so that makes it moot.

Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.

Interesting puzzle, but it hardly demonstrates a failure of rationality. The rational approach does not have to be the one that gets all the money, or even the bulk of it. In fact, the rational approach may in some cases be to stop thinking and consult a true random number generator to make your decision, thus rendering everyone's predictive powers useless. I don't think that's quite the case here though:

If your considering this after the alien has left, then you might as well stop, because its quite useless to do so. The alien is _gone_, and the boxes are already there and stuffed/not stuffed. Unless he comes back and says &quot;crap, screwed up, let me swap these real quick <img src='http://public.gamedev.net/public/style_emoticons/default/wink.gif' class='bbc_emoticon' alt='' /> <img src='http://public.gamedev.net/public/style_emoticons/default/wink.gif' class='bbc_emoticon' alt='' />&quot;, what you decide to do to the boxes has no effect on their contents. The only thing you control is whether or not to get $100. The fate of the $1,000,000 has already been determined, one way or the other. So open box B, and when its empty, don't worry about it; you're getting out a box of matches and lighting box A on fire first wouldn't have suddenly made $1,000,000 appear inside box B, especially since you've already opened box B. And if by chance box B does contain $1,000,000, then the alien was wrong, and your opening box A won't make that money suddenly disappear <img src='http://public.gamedev.net/public/style_emoticons/default/tongue.gif' class='bbc_emoticon' alt='' />.

Now if you make this decision BEFORE the alien makes his prediction, that's different. Now you can influence what the alien decides to do, and the $1,000,000 is back under your control. At this point the decision is pretty obvious: purchase a box cutter, some matches, and some ribs; then go outside and scream at the top of your lungs that &quot;You'll be cooking the alien some yummy BBQ with box A as fuel, just as soon as he shows up.&quot; Make sure to be polite, because #1. he's giving you $1,000,000, and #2. he is apparently so rich and powerful that he can fly around space setting up expensive logic-puzzles for random species.

Rationality is not the same as perfection and does not necessarily lead to optimal solutions at all times. A more interesting (and realistic) scenario is the Prisoner's dilemma, where rational behavior generally leads to suboptimal results.

I would think the OP's story is rather easy to solve: assuming the alien is speaking the truth, one choice gets you one million, the other one hundred. If you take into account probabilities, you get for B = 0.9999 * $1,000,000 + 0.0001 * $1,000,000= $1,000,000 and A+B = 0.9999 * $100 + 0.0001 * $1,000,100 = $200, hence a rational actor still picks B. The only irrational part I can see is in believing the alien can predict your future, but you assign percentages of probability to that, so that makes it moot.

Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.

The paradox then lies in the alien's ability to predict your choice and tell you about it. You can turn the tables on the alien by first opening B, and if the million dollars is in it, open A too, and you'll have $1,000,100. If B does not contain the million, don't open A. In both these cases, you have invalidated the alien's prediction.

A fun story about predictions is "Minority Report" (not the movie, but P.K. Dick's short story).

The problem with this riddle is that the decision has already been made.
It doesn't matter what you we're gonna do in this situation, since the decision has been made and the money has already been put in the boxes, you can safely take both boxes and feel happy you made some money.
The content of box B won't change depending on what you choose, you will simply realise of the results what kind of person you are, if you are greedy or just playing it safe.

Actually, the payoff is that once he has left (and cannot alter the results after he has left) you either get A+B or just B. Since A is always some positive number, a rational person would choose A+B. Remember, the boxes have already been stuffed.

That's not really how the payoff idea works. A payoff is the reward that you get for your action, and because the human in your story doesn't know whether or not B is empty (regardless of whether or not it's already been stuffed), a rational actor needs to assess the risk of choosing an empty B with the reward for choosing a stuffed B.

Choosing A is a safe choice, not necessarily a rational choice. The formula presented by Sam is a perfectly rational approach to the question, assuming that there aren't other factors that might affect the chooser's decision (like needing $100 immediately, for some reason).

There's a lot of behavioral research to suggest that when it comes to assessing risk humans (and all animals, really) are quite poor at behaving rationally-- that is, maximizing the benefit that they receive for the risk they take. There are certainle circumstances, as posted above, where people make non-optimal choices. But people are often irrational in making the choices they do, whether the outcome is optimal or not.

A fun riddle that I always liked is similar to yours, but the most interesting part to me is why people make the selection that they do:

You are presented with three boxes, and told that one box contains $100. The other two contain nothing. You are allowed to put your hand on one box, and then the presenter will take one of the other boxes away. The box that the presenter removes is guaranteed (and shown) to be an empty box. Then you are given a choice: You may stick with the box under your hand, and get whatever's inside, or switch to the other box. Which do you choose?

There is a rational choice to be made, but many people guess. Is rationality why you answered as you did?

The following assumes that the alien is not lying and is perfectly capable of predicting your actions.

My thinking is that if your decision path even allows you to take A, then box B will have nothing in it. If you make a plan that involves opening A, you will find B empty. If you accept that he has predicted your actions, then just take box B and the $1,000,000 it therefore must contain. If your plan includes taking both boxes, then the alien must have predicted it and you will "lose" overall.

It would seem that you if the alien's predictions are perfectly accurate, the rational player should discount the payoff of $1,000,100 as "impossible" because the alien would never allow this situation to arise. The only possibilities allowed are $1,000,000 or $100.

If you doubt the alien then all bets are off.

The paradox then lies in the alien's ability to predict your choice and tell you about it. You can turn the tables on the alien by first opening B, and if the million dollars is in it, open A too, and you'll have $1,000,100. If B does not contain the million, don't open A. In both these cases, you have invalidated the alien's prediction.

The alien has apparently predicted you'll take one or both. It doesn't allow you to walk away.